The next theorem shows that if Assumptions REF , REF , REF , and REF  are satisfied, then our cost complexity results derived for MLSVGD in Section  hold in the Bayesian inverse problem setting. These assumptions can be interpreted in the context of Bayesian inverse problems as follows: Assumption REF  and Assumption REF  are related to the forward model. Together they state that the approximation \(G_{\ell }\)  converges in an \(L^2\) -sense to \(G\)  as the level \(\ell \)  is increased. At the same time, as the level \(\ell \)  is increased and \(G_{\ell }\)  gets closer to \(G\) , the computational costs of evaluating \(G_{\ell }\)  may increase with a rate \(\gamma \) . This is typical behavior in, e.g., finite-element forward models where refining the mesh (increasing the level) leads to more accurate approximations and at the same time the computational costs of computing the finite-element solution increase with the number of mesh points. Furthermore, Assumption REF  is similar to the assumptions of [1]}, although there a pointwise bound is used. Assumption REF  is the convergence rate of SVGD and motivated by results from the literature as discussed in Section ; cf. Remark REF  for other convergence behavior. Assumption REF  ensures that the tail of the posterior distribution behaves as the tail of the prior and is similar to the envelope assumption made in, e.g., acceptance/rejection sampling [2]}.
